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| | <math>|(U^\circ \to X^\circ)| ~=~ |(\mathbb{B}^2 \to \mathbb{B}^2)| ~=~ 4^4 ~=~ 256.</math> | | | <math>|(U^\circ \to X^\circ)| ~=~ |(\mathbb{B}^2 \to \mathbb{B}^2)| ~=~ 4^4 ~=~ 256.</math> |
| |} | | |} |
| + | |
| + | Given any transformation of this type, <math>G : U^\circ \to X^\circ,</math> the (first order) |
| + | differential analysis of <math>G\!</math> is based on the definition of a couple of further transformations, derived by way of operators on <math>G,\!</math> that ply between the (first order) extended universes, <math>\operatorname{E}U^\circ = [u, v, du, dv]</math> and <math>\operatorname{E}X^\circ = [x, y, dx, dy],</math> of <math>G \operatorname{'s}\!</math> own source and target universes. |
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| <pre> | | <pre> |
− | Given any transformation of this type, G : U% -> X%, the (first order)
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− | differential analysis of G is based on the definition of a couple of
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− | further transformations, derived by way of operators on G, that ply
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− | between the (first order) extended universes, EU% = [u, v, du, dv]
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− | and EX% = [x, y, dx, dy], of G's own source and target universes.
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− |
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| First, the "enlargement map" (or the "secant transformation") | | First, the "enlargement map" (or the "secant transformation") |
| EG = <EG_1, EG_2> : EU% -> EX% is defined by the following | | EG = <EG_1, EG_2> : EU% -> EX% is defined by the following |